Inference in components of variance models with low replication

Abstract

In components of variance models the data are viewed as arising through a sum of two random variables, representing between- and within-group variation, respectively. The former is generally interpreted as a group effect, and the latter as error. It is assumed that these variables are stochastically independent and that the distributions of the group effect and the error do not vary from one instance to another. If each group effect can be replicated a large number of times, then standard methods can be used to estimate the distributions of both the group effect and the error. This cannot be achieved without replication, however. How feasible is distribution estimation if it is not possible to replicate prolifically? Can the distributions of random effects and errors be estimated consistently from a small number of replications of each of a large number of noisy group effects, for example, in a nonparametric setting? Often extensive replication is practically infeasible, in particular, if inherently small numbers of individuals exhibit any given group effect. Yet it is quite unclear how to conduct inference in this case. We show that inference is possible, even if the number of replications is as small as 2. Two methods are proposed, both based on Fourier inversion. One, which is substantially more computer intensive than the other, exhibits better performance in numerical experiments.